List of top Mathematics Questions on Continuity and differentiability asked in COMEDK UGET

If \[ y=\tan^{-1}\left( \frac{\sqrt{1+x^3}+\sqrt{1-x^3}} {\sqrt{1+x^3}-\sqrt{1-x^3}} \right) \] then \(\dfrac{dy}{dx}\) is equal to:
If \( 2y = \left[ \cot^{-1} \left( \frac{\sqrt{3} \cos x + \sin x}{\cos x - \sqrt{3} \sin x} \right) \right]^2 \ \quad \forall x \in \left( 0, \frac{\pi}{2} \right), \text{ then } \frac{dy}{dx} \text{ is equal to:}\)}
If \( f(x) = \begin{cases} \frac{1-x^m}{1-x}, & \text{for } x \neq 1 \\ 2m-1, & \text{for } x = 1 \end{cases} \) and the function is discontinuous at \( x = 1 \), then
If $y = | \cos\, x | + | \sin\, x |$, then $\frac{dy}{dx}$ at $x = \frac{2 \pi}{3}$ is
If $f(x) = \log_{x^2} (\log \,x)$, then $f'(x)$ at $x = e$ is
If $y = \tan \: x $ then $ \frac{d^2 y}{dx^2} $ =
If $x= \frac{1-t}{1+t} ; y= \frac{2t}{1+t}, $ then $\frac{d^{2}y}{dx^{2}} = $
If $f(x) = \begin{cases} \frac{e^{3x} - 1}{4x} & \quad \text{for} x \neq 0 \\ \frac{k + x}{4} & \quad \text{for } x= 0 \end{cases}$ is continuous at $x = 0$, then $k =$
If $y=\log \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) ,$ then $ \frac{dy}{dx} = $
If $y =\frac{\sec x +\tan x}{\sec x - \tan x} $ , then $\frac{dy}{dx} = $
If $\log y= m \tan^{-1} x,$ then
If $x= a\left(\cos \theta +\theta \sin \theta\right)$ and $ y = a \left(\sin \theta- \theta \cos \theta \right) $ where $0 < \theta < \frac{\pi}{2}$ , then $ \frac{d^{2}y}{dx^{2}}$ at $ \theta = \frac{\pi}{4}$ is equal to
Which of the following functions is differentiable at $x = 0$ ?
If $x= a\left(\cos t+\log \tan \frac{t}{2}\right), y = a \sin t, $ then $ \frac{dy}{dx} $
If $ y =\sin^{-1} \left(\frac{5x+12 \sqrt{1 -x^{2}}}{13}\right)$ , then $ \frac{dy}{dx} = $
If $u = f(x^2) , v = g(x^3) , f'(x) = \sin x $ and $g'(x) = \cos x,$ then $ \frac{du}{dv} = $
If $x= 3 \cos t - 2 \cos^{3} t , y = 3\sin t - 2 \sin^{3} t ,$ then $ \frac{d^{2}y}{dx^{2}} t = \frac{\pi}{6}$ is
If $\tan^{-1} \left(\frac{x}{y}\right) + \log \sqrt{x^{2} +y^{2}} = 0 $ , then $\frac{dx}{dy} = $
Let $f\left(x\right) = \frac{\log\left(1+ex\right)-\log\left(1-x\right)}{x} , x\ne0 $ . Then $f$ is continuous at $x = 0$ if $f(0)$ =
If $x^y = \log x$ , then $\frac{dy}{dx} $ at the point where the curve cuts the $x-axis$ is
The derivative of $\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) $ with respect to $\cot^{-1} \left(\frac{1-3x^{2}}{3x-x^{3}}\right)$ is
If $y = \tan^{-1} ( \sec x - \tan x) $, then $ \frac{dy}{dx} = $
If $\sqrt{1-x^{2} } + \sqrt{1- y^{2}} =x -y $, then $\frac{dy}{dx} = $
The function $f(x) = \begin{cases} x^2 & \quad \text{for } x < 1\\ 2 - x & \quad \text{for } x \geq 1 \end{cases}$ is